Data analysis device, method, and program

Abstract

An interval-valued matrix including elements represented by interval values can be resolved into factor matrices with high accuracy. A parameter estimation unit 20 estimates a factor matrix A and a factor matrix B such that an objective function, represented by including a probability of an element xij taking a scalar value thereof, which is represented using an estimate of the element xij estimated from the factor matrix A and the factor matrix B, for each element xij that is a scalar value, and a probability of the element xij taking an interval value thereof, which is represented using the estimate of the element xij estimated from the factor matrix A and the factor matrix B, for each element xij that is an interval value, is optimized.

1 . A data analysis device for resolving an interval-valued matrix X that is an I×J matrix having an element x ij  representing a relationship between a first object i (1≤i≤I, I is an integer equal to or greater than 1) and a second object j (1≤j≤J, J is an integer equal to or greater than 1), the element x ij  being a scalar value or an interval value, into an I×R factor matrix A having an element a ir  representing a relationship between the first object i and a factor r (1≤r≤R, R is an integer equal to or greater than 1) and a J×R factor matrix B having an element b jr  representing a relationship between the second object j and the factor r, the data analysis device comprising:
 a parameter estimator configured to estimate the factor matrix A and the factor matrix B such that an objective function is optimized, wherein the objective function includes:
 a probability of the element x ij  taking a scalar value thereof, which is represented using an estimate of the element x ij  estimated from the factor matrix A and the factor matrix B, for each element x ij  that is a scalar value, and 
 a probability of the element x ij  taking an interval value thereof, which is represented using the estimate of the element x ij  estimated from the factor matrix A and the factor matrix B, for each element x ij  that is an interval value. 
 
 
     
     
         2 . The data analysis device according to  claim 1 , wherein the probability of the element x ij  taking an interval value thereof includes a difference between:
 a cumulative density function representing a probability of the element x ij  taking a value equal to or less than an upper limit value of the interval value thereof, and   a cumulative density function representing a probability of the element x ij  taking a value equal to or less than a lower limit value of the interval value thereof.   
     
     
         3 . The data analysis device according to  claim 1 ,
 wherein the probability of the element x ij  taking a scalar value thereof is represented by a probability density function based on a normal distribution.   
     
     
         4 . The data analysis device according to  claim 1 , wherein the parameter estimator repeats update of the factor matrix A and the factor matrix B such that an auxiliary function that is an upper-bound function of the objective function decreases until predetermined repetition end conditions are satisfied. 
     
     
         5 . A data analysis method in a data analysis device for resolving an interval-valued matrix X that is an I×J matrix having an element x ij  representing a relationship between a first object i (1≤i≤I, I is an integer equal to or greater than 1) and a second object j (1≤j≤J, J is an integer equal to or greater than 1), the element x ij  being a scalar value or an interval value, into an I×R factor matrix A having an element a ir  representing a relationship between the first object i and a factor r (1≤r≤R, R is an integer equal to or greater than 1) and a J×R factor matrix B having an element b jr  representing a relationship between the second object j and the factor r, the data analysis method comprising:
 estimating, by a parameter estimator, the factor matrix A and the factor matrix B such that an objective function is optimized, wherein the objective function includes:
 a probability of the element x ij  taking a scalar value thereof, which is represented using an estimate of the element x ij  estimated from the factor matrix A and the factor matrix B, for each element x ij  that is a scalar value, and 
 a probability of the element x ij  taking an interval value thereof, which is represented using the estimate of the element x ij  estimated from the factor matrix A and the factor matrix B, for each element x ij  that is an interval value. 
 
 
     
     
         6 . The data analysis method according to  claim 5 , wherein the probability of the element x ij  taking an interval value thereof includes a difference between:
 a cumulative density function representing a probability of the element x ij  taking a value equal to or less than an upper limit value of the interval value thereof and   a cumulative density function representing a probability of the element x ij  taking a value equal to or less than a lower limit value of the interval value thereof.   
     
     
         7 . The data analysis method according to  claim 5 , wherein the estimating by the parameter estimator includes repeating update of the factor matrix A and the factor matrix B such that an auxiliary function that is an upper-bound function of the objective function decreases until predetermined repetition end conditions are satisfied. 
     
     
         8 . A program for causing a computer to serve as each component constituting the data analysis device for resolving an interval-valued matrix X that is an I×J matrix having an element xii representing a relationship between a first object i (1≤i≤I, I is an integer equal to or greater than 1) and a second object j (1≤j≤J, J is an integer equal to or greater than 1), the element x ij  being a scalar value or an interval value, into an I×R factor matrix A having an element a ir  representing a relationship between the first object i and a factor r (1≤r≤R, R is an integer equal to or greater than 1) and a J×R factor matrix B having an element b jr  representing a relationship between the second object j and the factor r, the data analysis device comprising:
 a parameter estimator configured to estimate the factor matrix A and the factor matrix B such that an objective function is optimized, wherein the objective function includes:
 a probability of the element x ij  taking a scalar value thereof, which is represented using an estimate of the element x ij  estimated from the factor matrix A and the factor matrix B, for each element x ij  that is a scalar value, and 
 a probability of the element x ij  taking an interval value thereof, which is represented using the estimate of the element x ij  estimated from the factor matrix A and the factor matrix B, for each element x ij  that is an interval value. 
 
 
     
     
         9 . The data analysis device according to  claim 2 , wherein the probability of the element x ij  taking a scalar value thereof is represented by a probability density function based on a normal distribution. 
     
     
         10 . The data analysis device according to  claim 2 , wherein the parameter estimator repeats update of the factor matrix A and the factor matrix B such that an auxiliary function that is an upper-bound function of the objective function decreases until predetermined repetition end conditions are satisfied. 
     
     
         11 . The data analysis device according to  claim 3 , wherein the parameter estimator repeats update of the factor matrix A and the factor matrix B such that an auxiliary function that is an upper-bound function of the objective function decreases until predetermined repetition end conditions are satisfied. 
     
     
         12 . The data analysis method according to  claim 6 , wherein the estimating by the parameter estimator includes repeating update of the factor matrix A and the factor matrix B such that an auxiliary function that is an upper-bound function of the objective function decreases until predetermined repetition end conditions are satisfied. 
     
     
         13 . The program according to  claim 8 , wherein the probability of the element x ij  taking an interval value thereof includes a difference between;
 a cumulative density function representing a probability of the element x ij  taking a value equal to or less than an upper limit value of the interval value thereof, and   a cumulative density function representing a probability of the element x ij  taking a value equal to or less than a lower limit value of the interval value thereof.   
     
     
         14 . The program according to  claim 8 , wherein the probability of the element x ij  taking a scalar value thereof is represented by a probability density function based on a normal distribution. 
     
     
         15 . The program according to  claim 13 , wherein the probability of the element x ij  taking a scalar value thereof is represented by a probability density function based on a normal distribution. 
     
     
         16 . The program according to  claim 8 , wherein the parameter estimator repeats update of the factor matrix A and the factor matrix B such that an auxiliary function that is an upper-bound function of the objective function decreases until predetermined repetition end conditions are satisfied. 
     
     
         17 . The program according to  claim 13 , wherein the parameter estimator repeats update of the factor matrix A and the factor matrix B such that an auxiliary function that is an upper-bound function of the objective function decreases until predetermined repetition end conditions are satisfied. 
     
     
         18 . The program according to  claim 14 , wherein the parameter estimator repeats update of the factor matrix A and the factor matrix B such that an auxiliary function that is an upper-bound function of the objective function decreases until predetermined repetition end conditions are satisfied. 
     
     
         19 . The program according to  claim 15 , wherein the parameter estimator repeats update of the factor matrix A and the factor matrix B such that an auxiliary function that is an upper-bound function of the objective function decreases until predetermined repetition end conditions are satisfied. 
     
     
         20 . The data analysis method according to  claim 6 , the method further comprising:
 generating, based on the estimated factor matrix A and factor matrix B, a latent pattern of data collected through questionnaires, wherein the data includes the element x ij .